Explorations in Perimeter and Area
Teacher’s Edition
Answers to questions are in blue and in italics.
Additional teaching comments are in black and labeled TC-#
Introduction
The goal of this lesson is to help students understand how changes in one or two
dimensions affect the area and perimeter of rectangles. Students will be directed to
generate a data chart from which they can discover the patterns that will lead to a deep
understanding of the relationship between perimeter and area. The National Council of
Teachers of Mathematics defines mathematics as, “The language and science of
patterns,” and it is through the study of patterns that students will be able to build their
understanding. It is imperative then that the teacher focus discussion and individual
directives on any patterns the students are generating.
Students can be grouped in any of several ways. Suggested grouping is to have groups
of four with each group having a heterogeneous mix of students. Each student will be
responsible for their own copy of the work and their own responses to many of the
questions, but the lesson as a whole is designed for students to work together and build
their understandings together through partner discussion, group discussion and whole
class discussion with each person’s evidence being their individual written responses.
The pretest is a tool to help the teacher discover what the students already know about
area. If the class has already completed some assignments or a unit on area the
pretest may not be necessary. It is up to the teacher’s professional judgment to decide
if the pretest is needed. All students need to know how to find the area of a square and
a rectangle before this lesson. The extensions to this lesson include other polygons
and prisms, but the ability to find the area of those regions is not essential to this lesson.
Most of the questions throughout the assignment are in-process questions. They
should not be graded as correct or in-correct as such, but as complete or incomplete.
As students explore new ideas it should be safe from concern abut being wrong.
NOTES:
• The use of calculators is encouraged as this problem is about the students
understanding of perimeter and area, not their ability to perform arithmetic.
• Some teachers have found it helpful to have the students use both a pencil and a
pen while working on problems like this. The student uses the pencil for all of
their own work and the pen is used when they get ideas from someone else.
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Problem: Explorations in Area and Perimeter
Group Names:
__________________________ (me)
__________________________
_________________________
__________________________
Introduction
Essential Question: When the dimensions of a geometrical figure change, how does
the area enclosed by that shape change?
Predict
1.
How does the area of a square change when you double the length of one
dimension (creating a rectangle)? (Estimates only, no calculations at this time.
Defend your answer with mathematical reasoning.)
Student’s answers will vary. This question is in part a pre-assessment for the teacher to
understand the student’s needs and mathematical background. (Possible student
answers include: 1) The area will double because the side doubled. 2) the area will go
up by 3 or 4 times {usually without much explanation}.
TC-1 - Clarify the question as necessary so the students understand their task. Give
them 1.5 to 2 minutes to write their prediction. While they are writing the teacher should
be circulating and encouraging students who are struggling to write something, even if it
is not correct. Make sure students know this question is not graded and that they
should be as clear as possible with their mathematical reasoning.
TC-2 - When the 1.5 to 2 minutes are up have the students share their results with their
partner. Give students 45 seconds per person to explain their thinking. Each student
should get their own time to talk without interruption from their partner. Coach the
students who are listening to just listen. Their job is only to understand their partner at
this time, not to correct them.
TC-3 - Spend another 3 to 5 minutes getting some responses from the class as a
whole. Again the point here is to focus on mathematical thinking, not to correct
anyone’s ideas. Elicit two or three ideas or questions from the class then indicate that
the next several problems should speak to these ideas and provide a mathematical
basis for deep understanding of how area changes when the dimensions of the polygon
change.
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TC-4 - Groups should then solve the next 4 problems making sure they fill out the data
chart as they go.
Explore
2.
Find the areas and perimeters of the following figures: (Label your answers)
A)
10 cm
Perimeter = _____
10 cm
Area = ______
B)
C)
30 cm
10 cm
Perimeter = _____
20 cm
Perimeter = _____
10 cm
Area = ______
Area = ______
TC-5 - As you circulate suggest that each person find all perimeters and all are as and
then check in with each other person in their group to correct their answers. All
students need to practice finding the areas.
TC-6 - For number 3 each student needs to fill out their own prediction and then each
group member should check in– every member reading their prediction to the group.
The group should then pick the prediction that seems to have the best mathematical
reasoning behind it. The group could be asked to share their prediction with the class
later on, so they will need to have it written down and be able to explain it.
TC-7 - If you notice a student is having trouble (or you could predict from the pre-test
that they would be struggling) then direct them to use the quarter inch graph paper and
draw each of the rectangles discussed. One square would equal one unit and each
rectangle could be counted for both perimeter and area.
Predict
3. Does it matter which dimension of the rectangle is doubled or tripled to create the
new shape when you are predicting a change in area? Explain your reasoning.
Student’s answers will vary. Emphasize the mathematical thinking – or logical reasoning
in their explanations. (Possible student answers include: Yes it matters, when the base
is doubled the figure is twice as big but when the height is doubled it doesn’t look nearly
as big.)
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Explore
4.
Find the areas and perimeters of the following figures: (Keep the length constant
and change only the width) (Label your answers)
A)
8 cm
C)
Perimeter = _____
8 cm
5 cm
Area = ______
Perimeter = _____
B)
15 cm
8 cm
10 cm
Area = ______
Perimeter = _____
Area = ______
5.
Find the areas and perimeters of the following figures: (Keep the width constant
and change only the length) (Label your answers)
8 cm
Perimeter = _____
5 cm
Area = ______
B)
16 cm
Perimeter = _____
5 cm
Area = ______
C)
24 cm
Perimeter = _____
5 cm
Area = ______
TC-8 - Check in with groups as they work…they should be showing their work
and filling out the data table as they go.
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Look for a pattern
6.
(Fill in the data table with your areas and perimeters)
When one dimension of a square or rectangle was doubled (or tripled) in length
how did the area change?
The area of the figure is doubled when one dimension is doubled, length or width.
7.
Is there a pattern to how the areas are increasing? If so explain the pattern.
Student’s answers will vary. (Possible student answers include: As the dimensions get
bigger, the area also gets bigger.)
TC-9 - Is this pattern predictable? Be ready to suggest other sizes of rectangles with
which students can test their patterns or conjectures.
TC-10 - Bring the class together here to present findings, methods and patterns found
so far. If you have noticed some groups that have made some decent progress then
call on them. (I often carry a four-sided die with me and pick the person to present from
the table by rolling the die. Each person at the table has a number before I roll the die.)
TC-11 - As students present ask clarifying questions like:
What did you notice that helped you to find that pattern? Can you use that pattern on a
different rectangle? Will it still be true? Or encourage members of the class to ask their
own questions to clarify the math being presented.
Predict
8.
When 2 dimensions (length and width) of a square or rectangle are doubled (or
tripled) how will the area change?
Student’s answers will vary. (Possible student answers include: The area will double
{this student is not acknowledging the change in two dimensions})
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Explore
9.
Find the areas and perimeters of the following squares: (Label your answers)
A)
10 cm
Perimeter = _____
10 cm
C)
Area = ______
20 cm
B)
30 cm
Perimeter = _____
30 cm
Area = ______
20 cm
Perimeter = _____
Area = ______
10.
Find the areas and perimeters of the following rectangles: (Label your answers)
A)
5 cm
8 cm
B)
16 cm
Perimeter = _____
Perimeter = _____
Area = ______
10 cm
Area = ______
C)
24 cm
Perimeter = _____
15 cm
Area = ______
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TC-12 - Ask groups as you circulate if they have noticed any patterns in the way the
perimeters are changing. Any connections you can help them make between the
perimeter (whether one dimension or two are being changed) and the resulting area will
be helpful. (For example, if one dimension is doubled the area is doubled, but if two
dimensions are doubled the area goes up by four times – double-double.)
TC-13 - The following section is very important. Students have a difficult time with
meta-cognition and the following questions ask them to examine their own thinking.
You may want to stop and give an example to demonstrate meta-cognition. This
example from your own thinking will help them to understand what types of answers are
expected from the following questions.
Look for a pattern
11. What patterns do you notice in how the areas are increasing? Examine the data
table and compare the changes in the length or width to the changes in the area.
When one side doubles the area doubles. When a dimension changes, that change is
multiplied by the other dimension and the result is the change in area. When both sides
of the rectangle are doubled the area increases by four times. When both sides triple
the area goes up by 9 times. When two dimensions are changed at the same time the
amount of change is the product of the changes to each of the individual sides.
12. Did you accurately predict the change in area when both dimensions were doubled?
_________________
13. If so explain why/how mathematically you were able to predict that result?
______________________________________________________________________
______________________________________________________________________
14. If your prediction was incorrect how did your answer compare to the actual result?
Too large
Too small
(Circle your choice)
15. Can you explain what was wrong with your thinking or your calculations that
resulted in your answer?
______________________________________________________________________
______________________________________________________________________
TC-14 - Direct students to examine each other’s work at this time focusing on the
mathematics (not whether it is ‘right’ or ‘wrong’). Each student should explain the
thinking they had that led to their answer – correct or incorrect. The goal is to have
another student understand their mathematical thinking and therefore help them find the
error in their thinking if there was one.
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TC-15 - Keep directing students to focus on finding patterns in their answers. They can
also be challenged to find patterns in the way the problems are presented which may
help them find patterns in their work.
TC-16 - The key to numbers 16 and 18 is the word ‘factor.’ During group and class
discussions the teacher should focus their questioning on drawing out from students the
fact that when one dimension is doubled (x2) the area is also doubled. Then when both
the length and the width are doubled (two x2’s) the area is then affected by both
changes therefore increasing by four times (x4).
Summarize
16. When one dimension of a rectangle is doubled by what factor does the area
change?
The area increases by a factor of 2._________________________________________
______________________________________________________________________
17. Describe the pattern (in detail) or give a mathematical reasoning to prove or support
your statement in # 16.
______________________________________________________________________
______________________________________________________________________
18. When two dimensions of a rectangle are doubled by what factor does the area
change? (See the data table focusing on #’s 9, 10)
The are increases by a factor of 4. _________________________________________
19. Describe in detail the pattern(s) that prove your statement in # 18.
______________________________________________________________________
______________________________________________________________________
Predict
20. If the rectangle (5x8) was expanded by 4 times the original size in both dimensions,
what would you expect for the change in area? Explain the pattern or mathematical
reasoning that leads you to your answer.
The area would increase by a factor of 16. When one dimension is doubled the area
doubles. When both sides are doubled the area increases by 4 times. And when both
sides are tripled the area increases by a factor of 9. Each time both dimensions are
changed the area changes by the product of those changes.
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Extension # 1
TC-17 - This extension is recommended for all students. If there are students that are
really struggling before this point they should be directed to the differentiation activities
instead. Otherwise, if time allows, continue through this first extension section.
Essential Question: All of the problems so far have dealt with doubling or tripling side
lengths of the given shape (square or rectangle). What would happen if 5 units were
added to one side (or the other side) instead of multiplying that side length by 2 or 3?
Predict -> Look at number 23 A) below for diagrams as you do numbers 21 and 22.
21. Think about the patterns you have seen so far and predict the change in area if 5
units are added to the length.
______________________________________________________________________
______________________________________________________________________
22. Will adding 5 units to the length create a new figure with the same area as adding 5
units to the width? (Give your prediction with mathematical reasoning.)
______________________________________________________________________
______________________________________________________________________
Explore
23. Find the areas and perimeters of the following rectangles: (Label all answers)
A)
12 cm
Perimeter = _____
9 cm
Area = ______
C)
17 cm (12 + 5)
B)
12 cm
Perimeter = _____
14 cm
(9 + 5) Area = ______
Perimeter = _____
9 cm
Area = ______
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Look for a pattern
24. Did your predictions match the results? _________________________________
25. The areas in the second two figures are not the same, explain why.
a. Look at figure 23 B) above; how much area is added to the figure when 1 cm is
added to the height? 2cm? 5cm?
In figure B) a one cm addition adds a row that is 12 cm in length. 2 cm adds two rows of
12 or 24 square cm. 5 cm therefore adds 5 * 12 or 60 square cm,
______________________________________________________________________
b. Look at part 23 C) above; how much area is added to the figure when 1cm is
added to the length? 2cm? 5cm?
In figure C) a one cm addition adds a column that is 9 cm in height. 2 cm adds two
columns of 9 or 18 square cm. 5 cm therefore adds 5 * 9 or 45 square cm.
______________________________________________________________________
c. Now can you explain why the rectangle did not have the same area when 5 cm
were added to the length as compared to when the 5 cm were added to the
width?
In figure B the area is increased by 5 rows of 12. That is a total increase of 60 square
units. In figure C the width is increased by 5, which adds 5 columns of 9 for an increase
of 45 square units.
26. If your predictions were wrong, explain what about your thinking led you to the
incorrect answer.
Student answers will vary.
TC-18 - Ask the students to explain this answer in their small groups. As you
circulate you can quickly check in with one person per table and ask them how
their partner answered that question, or if they can answer what their partner’s
mistake was.
27. To generalize from the above work, when one cm (or _n_cm) is added to the length
of a rectangle, then ___1 * the width or n * the width ___is added to the rectangle’s
area.
Analyze
28. Examine number 23 carefully. How is the change in area related to the change in
side length?
The area change is related to the side length by the fact that any one unit of length gain
on the side dimension adds a column to the area of the figure. That column is as long
as the other dimension of the rectangle.
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29.
Examine one more situation. Compute the areas and perimeter of the following
figure. This figure is the next extension of the three in number 23. (Label each
dimension of the rectangle and find each sub-area):
17 cm (12 + 5)
Perimeter = __62_
14 cm
(9 + 5)
108
45
60
25
Area = _238__
30. Describe how the patterns you found in other problems relate to this situation where
5 units were added to the width, the 5 units were added to the length and finally (#
29) 5 units were added to both the length and the width.
______________________________________________________________________
______________________________________________________________________
31. In general when two sides of a rectangle are doubled the area increases by ___4__
times because __
Doubling one side doubles the area of the rectangle. Doubling the second side also
doubles the area. When the area has been doubled by the first side and then is
doubled again that increases the area by a factor of 4.
32. Also if two sides of a rectangle are tripled (or increased by any factor) then what is
the resulting change in area?
Using the same logic as in number 31 the total increase will be 9 times the original
area._
TC-19 – If students are having struggles with this concept I will have them draw
different rectangles and then follow the instructions for drawing the next rectangle
and then look at the areas and to see how they add up (similar to the diagram for
number 29).
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Extension # 2
TC-20 - Students that pursue these problems should be the ones that did not
experience difficulties in the first section.
Predict
Do the patterns you found hold true for other geometrical shapes? Include both other 2dimensional shapes (like circles or hexagons) and 3-dimensional shapes like
rectangular prisms and pyramids) In your prediction be as clear as you can with your
mathematical reasoning.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
On the additional lab sheets provided are shapes to work with for more extension
problems.
33. How does this pattern work for other shapes?
• Triangles
• Circles
• Rectangles with unusual side measurements
• Other polygons (your choice)
34. How does this pattern extend into three dimensions? (Volume)
• Cubes
• Rectangular prisms
• Pyramids
• Spheres
• Cones
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Record all data in the following chart:
Problem
Length
Width
Perimeter
#
Area
Pattern
Problem
#
Lengt
h
Width
Perimeter
Area
Pattern
2A
10
10
40
100
A
9A
10
10
40
100
A
2B
10
20
60
200
2A
9B
20
20
80
400
4A
2C
10
30
80
300
3A
9C
30
30
120
900
9A
5
8
26
40
A
5
5
20
25
A
4B
10
8
36
80
2A
10
10
40
100
4A
4C
15
8
46
120
3A
15
15
60
225
9A
5
8
26
40
A
23A
9
12
42
108
A
5B
5
16
42
80
2A
23B
14
12
52
168
A+(5*12)
5C
5
24
58
120
3A
23C
9
17
52
153
A+(5*9)
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4A
5A
Explorations in Perimeter and Area
10A
10B
10C
Problem
#
Data Table
for Extension
#2 Shapes:
Length
Area
8
34.4
48
2A
8
15
3A
16
68.4
192
Pi
50.3
201.1
Pi
100.5
804.2
9
48
135
2A x 4
16
2B
3A x4
30
18
96
540
Shape number
Length
Width
Height
Area of Base
4A
14
7
10
98
4B
28
14
20
392
5A
5B
6A
6B
7A
7B
Explorations in Perimeter and Area
Pattern
1A x 4
24
1B
3B
Perimeter
12
1A
This table is for extension
#2 problems which
investigate the change in
volume when there is a
change in one, two or three
dimensions of the original
prism.
Width
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Volume
980
7840
The volume
increases by 8
times
Extension
#1-B
Extension
#1-A
8 cm
16 cm
12 cm
24 cm
Extension
#2-A
Extension
#2-B
Radius = 16 cm
Radius = 8 cm
Extension
#3-A
Extension
#3-B
9 cm
15 cm
18 cm
30 cm
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Extension
#4-A
Extension
#4-B
Length = 14 cm
Width = 7 cm
Height = 10 cm
Length = 28 cm
Width = 14 cm
Height = 20 cm
Extension
#5-A
Explorations in Perimeter and Area
Extension
#5-B
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Extension
#6-A
Extension
#6-B
Extension
#7-A
Explorations in Perimeter and Area
Extension
#7-B
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